Parameters CSTR modeling

Needless to mention that the Danckwerts model [Eqs. (146)-(147)] reduces to the ideal PFR and the ideal CSTR models in the limits of Pe —> oo and Pe —> 0, respectively. As seen in Fig. 9, the ideal PFR model and the ideal CSTR model bound the solution of the Danckwerts model [Eqs. (146)—(147)] for the case of any finite Pe number. As shown in Fig. 9 (and from Eq. (150)), for Pe 10, the two-mode model predicts very close to the Danckwerts model when the variances of the two models are matched using Eqs. (71) and (63). However, the two-mode models can predict conversions in regions in the parameter space which are inaccessible to the Danckwerts models. These are the micromixing limited conversions which are even below conversions predicted by the ideal... 

For the CSTR model, the highest intersection point is physically realistic and, indeed, important. It corresponds to an intersection point on the deceleratory part of the reaction rate curve as indicated in Fig. 5.6(a, b). There are now two possible tangencies of the heat loss line with the heat release curve as some parameter such as the residence time is varied, as indicated in the figure. The first has a similar implication as in the Semenov case. It corresponds to the merger of two low-lying steady-states and to an ignition point on the steady-state locus and, in this model, arises typically as the residence time is increased. The system now jumps to a high steady-... 

Figure 14-18(a) describes a real PFR or PER with channeling that is modeled as two PFRs/PBRs in parallel. The two parameters are the fraction of flow 10 the reactors [i.e., (3 and (1 - p)] and the fractional volume [i.e.. a and (1 - Qf] of each reactor. Figure 14-18(b) describes a real PFR/PBR that has a backmix region and is modeled as a PFR/PBR in parallel with a CSTR. Figures H-19(a) and (b) show a real CSTR modeled as two CSTRs with interchange. In one case, the fluid exits from the top CSTR (a) and in the other case the fluid exits from the bottom CSTR. The parameter p represents the interchange volumetric flow rate and a the fractional volume of the top reactor, where the fluid exits the reaction system. We note that the reactor in model 14-19(b) was found to describe extremely well a real reactor used in the production of terephthalic acid. A number of other combinations of ideal reactions can be found in Levenspiel. ... 

For the ideal plug flow, tank in series and ideal CSTR models, k i the parameter to which conversion is most sensitive.. ... 

The microkinetic analysis of the H2 TPD data from Cu catalysts revealed that the influence of readsorption was negligible rendering the mathematically easier CSTR model appropriate. In this case, the proper kinetic desorption parameters can be extracted from the shifting TPD peak maximum temepratures as function of the heating rate. However, the example illustrated that deriving the coverage dependence from single crystal data is not without pitfalls. 

The consequences of insufficient mixing cannot be assessed with the CSTR model. They require the space dependence of process parameters like concentration and temperature to be accounted for. This may be done by a model on the basis of Computational Fluid Dynamics (CFD). 

The dimensionless groups involved in the solution are different from those of Ugelstad for two reasons. First, the additional variable of reactor mean residence time, 0, leads to one more dimensionless group, 3. Second, the particle volume parameter used in the dimensionless groups is the volume of the swollen seed particle. This volume is preferred to the average effluent volume because the seed characteristics are known before the problem is solved. Table 1 gives a list of the dimensionless parameters, with the defining equations, which are used in the CSTR model development. 

One of the interesting, and perhaps unexpected, results of the CSTR model studies is the nature of predicted particle size distributions. Figure 6 shows the influence of radical desorption from the latex particles (the parameter y) on the effluent latex PSD. The distributions are plotted in terms of a dimensionless diameter d/(6v /tt) /. ... 

Ratto, M., Paladino, O. Analysis of controlled CSTR models with fluctuating parameters and uncertain parameters. Chem. Eng. J. 79, 13-21 (2000)... 

Real reactors deviate more or less from these ideal behaviors. Deviations may be detected with re.sidence time distributions (RTD) obtained with the aid of tracer tests. In other cases a mechanism may be postulated and its parameters checked against test data. The commonest models are combinations of CSTRs and PFRs in series and/or parallel. Thus, a stirred tank may be assumed completely mixed in the vicinity of the impeller and in plug flow near the outlet. 

Distribution models are curvefits of empirical RTDs. The Gaussian distribution is a one-parameter function based on the statistical rule with that name. The Erlang and gamma models are based on the concept of the multistage CSTR. RTD curves often can be well fitted by ratios of polynomials of the time. 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

The described experimental rig for the anionic polymerisation of dienes has been shown to behave as an ideal CSTR. The mathematical model developed allows the prediction of the MWD at future points in the reactor history, once suitable kinetic parameters have been estimated. 

The goal is to determine a functional form for (a, b,. .., T) that can be used to design reactors. The simplest case is to suppose that the reaction rate has been measured at various values a,b,..., T. A CSTR can be used for these measurements as discussed in Section 7.1.2. Suppose J data points have been measured. The jXh point in the data is denoted as S/t-data aj,bj,..., Tj) where Uj, bj,..., 7 are experimentally observed values. Corresponding to this measured reaction rate will be a predicted rate, modeii p bj,7 ). The predicted rate depends on the parameters of the model e.g., on k,m,n,r,s,... in Equation (7.4) and these parameters are chosen to obtain the best fit of the experimental... 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

Different reactor networks can give rise to the same residence time distribution function. For example, a CSTR characterized by a space time Tj followed by a PFR characterized by a space time t2 has an F(t) curve that is identical to that of these two reactors operated in the reverse order. Consequently, the F(t) curve alone is not sufficient, in general, to permit one to determine the conversion in a nonideal reactor. As a result, several mathematical models of reactor performance have been developed to provide estimates of the conversion levels in nonideal reactors. These models vary in their degree of complexity and range of applicability. In this textbook we will confine the discussion to models in which a single parameter is used to characterize the nonideal flow pattern. Multiparameter models have been developed for handling more complex situations (e.g., that which prevails in a fluidized bed reactor), but these are beyond the scope of this textbook. [See Levenspiel (2) and Himmelblau and Bischoff (4).]... 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

Stirred tank performance often is nearly ideal CSTR or the model may need to take into account bypassing, stagnant zones or other parameters associated with the geometry and operation of the vessel and the agitator. Sometimes the vessel can be visualized as a zone of complete mixing in the vicinity of the impellers followed by a plug flow zone elsewhere, thus a CSTR followed by a PFR. 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as... 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

The family of curves represented by eqn. (46) is shown in Fig. 11 and the mean and variance of both the E(f) and E(0) RTDs are as indicated in Table 5. When N assumes the value of 0, the model represents a system with complete bypassing, whilst with N equal to unity, the model reduces to a single CSTR. As N continues to increase, the spread of the E 0) curves reduces and the curve maxima, which occur when 0 = 1 —(1/N), move towards the mean value of unity. When N tends to infinity, E(0) is a dirac delta function at 0 = 1, this being the RTD of an ideal PER. The maximum value of E(0), the time at which it occurs, or any other appropriate curve property, enables the parameter N to be chosen so that the model adequately describes an experimental RTD which has been expressed in terms of dimensionless time see, for example. Sect. 66 of ref. 26 for appropriate relationships. 

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